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Marcinkiewicz's interpolation theorem for Hardy-type spaces and its applications V. G. Krotov Faculty of Mechanics and Mathematics, Belarusian State University, Minsk, Belarus
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   § 1. Introduction1.1. The main notationLet $(X,\mu)$ be a set with $\sigma$-finite measure $\mu$ and $L^0(X)$ be the set (of equivalence classes) of measurable complex-valued functions on $X$. Consider the functional
$$
\begin{equation*}
\|f\|_{L^0(X)}:=\inf_{\delta>0}\bigl\{\mu\{|f|>\delta\}+\delta\bigr\}, \qquad f\in L^0(X).
\end{equation*}
\notag
$$
This functional defines a metric on $L^0(X)$ by
$$
\begin{equation}
d_{L^0}(f_1,f_2):=\|f_1-f_2\|_{L^0(X)}, \qquad f_1,f_2\in L^0(X).
\end{equation}
\tag{1.1}
$$
Convergence in this metric coincides with convergence in measure on $X$. For $0<p<\infty$ we let $L^p(X)$ denote the subset of $L^0(X)$ consisting of functions with finite quasinorm
$$
\begin{equation*}
\|f\|_{L^p(X)}:=\biggl(\int_X|f|^p\,d\mu\biggr)^{1/p}.
\end{equation*}
\notag
$$
Next, let $L^\infty(X)$ be the subset of $L^0(X)$ consisting of essentially bounded functions. We also set
$$
\begin{equation*}
\|f\|_{L^\infty(X)}:=\inf\bigl\{\lambda\colon \mu\{x\in x\colon |f(x)|>\lambda\}\bigr\}=0.
\end{equation*}
\notag
$$
We denote by $L^{p_0}(X)+L^{p_1}(X)$ the space of functions $f\in L^0(X)$ representable in the form $f=f_0+f_1$, where $f_0\in L^{p_0}(X)$ and $f_1\in L^{p_1}(X)$. We also set
$$
\begin{equation*}
\|f\|_{L^{p_0}(X)+L^{p_1}(X)}:=\inf\bigl\{\|f_0\|_{L^{p_0}(X)}+\|f_1\|_{L^{p_1}(X)}\colon f=f_0+f_1\bigr\}.
\end{equation*}
\notag
$$
Let $(Y,\nu)$ be another set with $\sigma$-finite measure $\nu$, and let $T$ be an operator with values in $L^0(Y)$ defined on some subset of $L^0(X)$. The operator $T$ is called quasi-subadditive if there exists a positive number ${K_1=K_1(T)}$ such that1 If, in addition, then $T$ is quasilinear ($T(u+v)$ is assumed to be defined whenever $Tu$ and $Tv$ are; $T(\lambda u)$ is defined whenever $Tu$ is). In the case when $K_1=1$ we use the terms ‘subadditive’ and ‘sublinear’, respectively.We write $A\lesssim B$ if $A\leqslant CB$ for some positive constant $C$, which can depend on some parameters, which will be indicated explicitly. 1.2. Marcinkiewicz’s theoremIn the original version of Marcinkiewicz’s interpolation theorem for $L^p$-spaces, the linear or positive subadditive operator $T$ was assumed to satisfy a weak-type condition of the form
$$
\begin{equation}
\nu(\{|Tf|>\lambda\})\leqslant \biggl(\frac{M}{\lambda}\|f\|_{L^p(X)}\biggr)^q, \qquad \lambda>0, \quad f\in L^p(X)
\end{equation}
\tag{1.4}
$$
(see [1], [2] and also [3], Ch. 12, § 4, and [4], Appendix B). For a more general approach to results of this kind, see, for example, [5] and [6]. Condition (1.4) can be written as
$$
\begin{equation}
\|Tf\|_{L^{q,\infty}(Y)}\leqslant M\|f\|_{L^{p}(X)}, \qquad f\in L^{p}(X),
\end{equation}
\tag{1.5}
$$
where
$$
\begin{equation}
\|f\|_{L^{p,\infty}(X)}:=\sup_{\lambda>0}\lambda[\mu(\{|f|>\lambda\})]^{1/p}, \qquad p>0,
\end{equation}
\tag{1.6}
$$
is the weak $L^p$-quasinorm. Theorem 1. Let $T\colon L^{p_0}(X)+L^{p_1}(X)\to L^0(X)$ be a quasi-subadditive operator, and let $1\leqslant p_0\leqslant q_0\leqslant\infty$, $1\leqslant p_1\leqslant q_1\leqslant\infty$, $p_0<p_1$ and $q_0\ne q_1$. Assume that there exist constants $M_0$ and $M_1$ such that
$$
\begin{equation*}
\|Tf\|_{L^{q_0,\infty}(Y)}\leqslant M_0\|f\|_{L^{p_0}(X)}, \qquad f\in L^{p_0}(X),
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\|Tf\|_{L^{q_1,\infty}(Y)}\leqslant M_1\|f\|_{L^{p_1}(X)}, \qquad f\in L^{p_1}(X).
\end{equation*}
\notag
$$
Let $\theta\in(0,1)$,
$$
\begin{equation}
\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}\quad\textit{and} \quad\frac{1}{q}=\frac{1-\theta}{q_0}+\frac{\theta}{q_1}.
\end{equation}
\tag{1.7}
$$
Then ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$ and $\theta$).This result is commonly referred to as Marcinkiewicz’s interpolation theorem; this is an important tool in many branches of modern analysis involving $L^p$-spaces of summable functions. The first step towards Theorem 1 was the note [1] by Marcinkiewicz, which puts forward, without proof, its ‘diagonal’ variant $p_0=q_0$, $p_1=q_1$. The complete version of the result was later established by Zygmund [2]. 1.3. The Stein–Weiss–Grafakos theoremAn application of condition (1.5) to the characteristic function $\chi_A$ of a measurable set $A\subset X$ produces the inequality
$$
\begin{equation}
\|T\chi_A\|_{L^{q,\infty}(Y)}\leqslant M[\mu(A)]^{1/p},
\end{equation}
\tag{1.8}
$$
which is called a (restricted) weak-type inequality. In Marcinkiewicz’s interpolation theorem such conditions were originally introduced by Stein and Weiss [7] (see also § 5.3 in [8]), who assumed in [7] that $p_0,p_1,q_0,q_1\geqslant 1$ and considered the operators defined on the class $S(X)$ of simple functions on $X$ (finite linear combinations of characteristic functions of measurable sets of finite measure). Grafakos (see [9], Theorem 1.4.19) extended Marcinkiewicz’s interpolation theorem with the Stein–Weiss conditions (1.8) to arbitrary positive parameters $p$ and $q$. To formulate his result we recall the definition of Lorentz spaces. For $0<p,r\leqslant\infty$ we let $L^{p,r}(X)$ denote the Lorentz space (see [10] and also [8], § 5.3) with the quasinorm
$$
\begin{equation}
\|f\|_{L^{p,r}(X)}:= \begin{cases} \displaystyle \biggl(\int_0^{\infty}[t^{1/p}f^*(t)]^r\,\frac{dt}{t}\biggr)^{1/r},& 0<r<\infty, \\ \displaystyle \sup_{t>0}t^{1/p}f^*(t), & r=\infty, \end{cases}
\end{equation}
\tag{1.9}
$$
where $f^*$ is the decreasing equimeasurable rearrangement of the function $f$ on $X$:
$$
\begin{equation*}
f^*(t):=\inf\bigl\{s>0\colon \mu(\{|f|>s\})\leqslant t\bigr\}, \qquad t>0,
\end{equation*}
\notag
$$
(for example, see [8], § 5.3). For $r=\infty$ the quasinorm (1.9) coincides with (1.6) (see [8], Lemma 3.8). Theorem 2. Let $0<p_0\ne p_1\leqslant\infty$, $0<q_0\ne q_1\leqslant\infty$ and $0<r<\infty$, and let $T\colon L^{p_0}(X)+L^{p_1}(X)\to L^0(Y)$ be a quasilinear continuous operator satisfying the following condition: there exist constants $M_0$ and $M_1$ such that, for each measurable set $A\subset X$ of finite measure and Let $\theta\in(0,1)$, and let $p$ and $q$ be defined by (1.7). Then ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$, $r$ and $\theta$).Theorem 1.4.19 in [9] is close to Theorem 2, but there the operator $T$ is not assumed to be continuous. However, the proof in [9] contained a gap, which was filled in [11], where both the results and proofs were corrected. The validity of Theorem 2 was pointed out in [11] (see Remark 1.4, (iii), (iv)), where its variant for operators defined on some subclass of the class $S(X)$ of simple functions on $X$ was also proved (Theorem 1.1 in [11]). The proof there is similar to the proof of Theorem 1.4.19 in [9]. A related version of the same theorem was later established by Grafakos [12] (see Theorem 1.4.19 there). The condition of local continuity in related problems (a multilinear version of Marcinkiewicz’s theorem) was used in [13]. Our aim here is to obtain analogues of Theorem 2 for operators satisfying weak-type $L^p$-inequalities and acting on some spaces which are natural extensions of the Hardy classes $H^p$ of analytic or harmonic functions. Some applications of these results will also be given. § 2. The main results2.1. The spaces $\mathcal H^p(\mathbf X)$Let $X$ be a Hausdorff space whose topology is induced by a quasimetric $d$, that is, $d\colon X\times X\to[0,\infty)$ is a function satisfying all axioms of a metric except the triangle inequality, which is replaced by a weaker condition: there exists a number $K_2=K_2(d)\geqslant 1$ such that, for all $x,y,z\in X$, Let $X$ also be equipped with a $\sigma$-finite Borel measure $\mu$ such that the measure of each ball
$$
\begin{equation*}
B(x,t):=\{y\in X\colon d(x,y)<t\}, \qquad x\in X, \quad t>0,
\end{equation*}
\notag
$$
is finite and positive. Consider the product
$$
\begin{equation}
\mathbf X:=X\times I, \quad \text{where } I=(0,t_0), \quad 0<t_0\leqslant+\infty,
\end{equation}
\tag{2.1}
$$
and $X$ is interpreted as the boundary of $\mathbf X$. We equip $\mathbf X$ with the standard product measure $\mu\times m_1$, where $m_1$ is the one-dimensional Lebesgue measure on $I$ (see [14], § 3.3). We consider the ‘nontangential’ domains
$$
\begin{equation}
D(x):=\{(y,t)\in\mathbf X\colon d(x,y)<t\}, \qquad x\in X,
\end{equation}
\tag{2.2}
$$
which approach points $x\in X$ of the ‘boundary’ of $\mathbf X$, and define the corresponding maximal function
$$
\begin{equation*}
\mathcal N u(x):=\sup\{|u(y,t)|\colon (y,t)\in D(x)\}, \qquad x\in X,
\end{equation*}
\notag
$$
for each function $u\colon \mathbf X\to\mathbb{C}$. We let $\mathcal H^0(\mathbf X)$ denote the set of all measurable functions $u\colon \mathbf X\to\mathbb{C}$ whose maximal function $\mathcal N u$ is finite $\mu$-almost everywhere (equivalent functions are not identified). We equip $\mathcal H^0(\mathbf X)$ with the metric
$$
\begin{equation*}
d_{\mathcal H^0(\mathbf X)}(u_1,u_2):=\|\mathcal N(u_1-u_2)\|_{L^0(X)}
\end{equation*}
\notag
$$
(see (1.1)). The convergence of $u_n$ to $u$ in this metric means that $\mathcal N(u_n-u)$ converges to zero in measure on $X$. Next, given $p,r>0$, let $\mathcal H^{p,r}(\mathbf X)$ be the class of functions $u\in\mathcal H^0(\mathbf X)$ such that
$$
\begin{equation*}
\|u\|_{\mathcal H^{p,r}(\mathbf X)}:=\|\mathcal N u\|_{L^{p,r}(X)} <\infty.
\end{equation*}
\notag
$$
For $r=p$ we write $\mathcal H^p(\mathbf X)$ in place of $\mathcal H^{p,p}(\mathbf X)$. In the case $\mathbf X=\mathbb R^{n+1}_+$ the classes $\mathcal H^p(\mathbb R^{n+1}_+)$ were considered for the first time in [15], where it was assumed in addition that functions in these classes are continuous and have nontangential limits almost everywhere (see also [16] and [17], where the case of general $\mathbf X$ was considered). The concept of a maximal function and the condition $\mathcal N u\in L^p(X)$, which date back to Hardy and Littlewood [18], are widely useful in the theory of Hardy spaces (see, for example, [19] for $\mathbf X=\mathbb R_+^{n+1}$, and Theorem 5.6.5 in [20], where $\mathbf X$ is the unit ball in $\mathbb{C}^n$). A number of boundary-value problems are formulated in terms of maximal functions (see, for example, [21] and [22]; this list is far from complete). This comment describes the possible range of application of our results. 2.2. The main resultsBy $\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)$ we denote the space of all functions $u\in \mathcal H^0(\mathbf X)$ representable as $u=u_0+u_1$, where $u_0\in \mathcal H^{p_0}(\mathbf X)$ and $u_1\in \mathcal H^{p_1}(\mathbf X)$. We equip this space with the quasinorm
$$
\begin{equation*}
\|u\|_{\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(y\mathbf X)}:=\inf\{\|u_0\|_{\mathcal H^{p_0}(\mathbf X)}+\|u_1\|_{\mathcal H^{p_1}(\mathbf X)}\colon u=u_0+u_1\}.
\end{equation*}
\notag
$$
It is easily seen that if $0<p_0<p<p_1$, then
$$
\begin{equation*}
\mathcal H^{p}(\mathbf X)\subset\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)
\end{equation*}
\notag
$$
and this embedding is continuous. The following important notation
$$
\begin{equation}
\widehat{A}:=\{x\in X\colon D(x)\cap A\ne\varnothing\}, \qquad A\subset\mathbf X,
\end{equation}
\tag{2.3}
$$
is used in our main results, which are as follows. Theorem 3. Let $0<p_0\ne p_1\leqslant\infty$, $0<q_0\ne q_1\leqslant\infty$ and $0<r<\infty$, and let $T$ be a quasilinear continuous operator from $\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)$ to $L^0(Y)$ which satisfies the following conditions: there exist positive constants $M_0$ and $M_1$ such that, for each measurable set $A\subset\mathbf X$ satisfying $\mu(\widehat{A})<\infty$, and Let $\theta\in(0,1)$, and let $p$ and $q$ be given by (1.7). Then for each function $u\in\mathcal H^{p,r}(\mathbf X)$, ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$, $r$ and $\theta$).Theorem 3 becomes Theorem 2 if $\mathbf X$, $\mathcal H^{p}(\mathbf X)$ and $\widehat{A}$ are replaced by $X$, $L^p(X)$ and $A\subset X$, respectively. Corollary 1. Under the assumptions of Theorem 3, let $\theta\in(0,1)$, and let $p$ and $q$, $p\leqslant q$, be defined by (1.7). Then for each function $u\in\mathcal H^{p}(\mathbf X)$, ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$ and $\theta$).Corollary 1 follows from Theorem 3 for $r=q$ since the scale of Lorentz spaces is monotone (see (3.3)). Corollary 2. Let the conditions of Theorem 3 be met. Then for each function $u\in\mathcal H^{p}(\mathbf X)$, ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$, $\theta$).Corollary 2 is a consequence of Theorem 3 with $r=p$. The next theorem is an analogue of the classical form of Marcinkiewicz’s theorem (see Theorem 1). Theorem 4. Let $0<p_0\ne p_1\leqslant\infty$, $0<q_0\ne q_1\leqslant\infty$, and let the quasilinear operator $T\colon \mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)\to L^ 0(Y)$ satisfy the following condition: there exist positive constants $M_0$ and $M_1$ such that, for all $\lambda>0$, and Let $\theta\in(0,1)$, and let $p$ and $q$, $p\leqslant q$, be defined by (1.7). Then for each function $u\in \mathcal H^p(\mathbf X)$, ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$, $\theta$).Now we state an analogue of Theorem 3 for operators acting from ${\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)}$ to $\mathcal H^0(\mathbf Y)$. Here $\mathbf Y=Y\times(0,\tau_0)$, where $(Y,\nu)$ is as above and $0<\tau_0\leqslant\infty$. In addition, $Y$ is equipped with a quasimetric $d_Y$ generating the maximal function
$$
\begin{equation*}
\mathcal N_Yu(y):=\sup\{|u(z,t)|\colon d_Y(y,z)<t\}, \qquad y\in Y,
\end{equation*}
\notag
$$
which, in its turn, generates the space $\mathcal H^0(\mathbf Y)$. Theorem 5. Let $0<p_0\ne p_1\leqslant\infty$, $0<q_0\ne q_1\leqslant\infty$ and $0<r<\infty$, and let $T$ be a quasilinear continuous operator from $\mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)$ to $\mathcal H^0(\mathbf Y)$ satisfying the following condition: there exist positive constants $M_0$ and $M_1$ such that for each measurable set $A\subset\mathbf X$ such that $\mu(\widehat{A})<\infty$, and Let $\theta\in(0,1)$, and let $p$ and $q$ be given by (1.7). Then for each function $u\in\mathcal H^{p,r}(\mathbf X)$ ($\lesssim$ depends on $K_1$, $p_0$, $p_1$, $q_0$, $q_1$, $r$ and $\theta$).Theorems 3–5 were announced by this author in [23]. For the proof of other results in [23], see [24]. § 3. Auxiliary results3.1. Hardy inequalitiesThe following inequalities are due to Hardy (see, for example, Appendix A in [4]): if $0<b<\infty$ and $1\leqslant p<\infty$, then
$$
\begin{equation}
\biggl(\int_0^{\infty}\biggl[\int_0^t|f(s)|\,ds\biggr]^pt^{-b-1}\,dt\biggr)^{1/p} \leqslant\frac{p}{b}\biggl(\int_0^{\infty}|f(t)|^pt^{p-b-1}\,dt\biggr)^{1/p}
\end{equation}
\tag{3.1}
$$
and
$$
\begin{equation}
\biggl(\int_0^{\infty}\biggl[\int_t^\infty|f(s)|\,ds\biggr]^pt^{b-1}\,dt\biggr)^{1/p} \leqslant\frac{p}{b}\biggl(\int_0^{\infty}|f(t)|^pt^{p+b-1}\,dt\biggr)^{1/p}.
\end{equation}
\tag{3.2}
$$
3.2. Lorentz spacesWe need the following properties of Lorentz spaces, which were defined in § 1.3 above. The scale of Lorentz spaces is monotone in the second parameter (see, for example, [9], Proposition 1.4.10, or [8], Theorem 3.11): if $0<p\leqslant\infty$ and $0< q_0<q_1\leqslant\infty$, then $L^{p,q_0}(X)\subset L^{p,q_1}(X)$ and ($\lesssim$ depends only on $p$, $q_0$ and $q_1$).For $p>1$ the quasinormed Lorentz space $L^{p,\infty}(X)$ is a normed space (see [25] or [9], Exercise 1.1.12): the corresponding norm
$$
\begin{equation*}
\|f\|_{L^{p,\infty}(X)}^*:=\sup[\mu(A)]^{-1+1/p}\int_A|f|\,d\mu
\end{equation*}
\notag
$$
(the supremum is taken over all measurable sets $A\subset X$ such that $0<\mu(A)<\infty$) satisfies
$$
\begin{equation}
\|f\|_{L^{p,\infty}(X)}\leqslant\|f\|_{L^{p,\infty}(X)}^*\leqslant\frac{p}{p-1}\|f\|_{L^{p,\infty}(X)}.
\end{equation}
\tag{3.4}
$$
The following two easy facts are worth mentioning (see (1.9)): for all $0<p,r<\infty$ and $0<q\leqslant\infty$, and
$$
\begin{equation}
\sum_{k\in\mathbb Z}[f^*(2^k)]^{\alpha}2^{k\alpha/p}\leqslant\frac{2^{\alpha/p}}{\ln2}\|f\|_{L^{p,\alpha}(X)}^{\alpha}.
\end{equation}
\tag{3.6}
$$
3.3. The Aoki–Rolewicz lemmaFor the proof of the following result, see [26], [27] and also [12], Exercise 1.4.6. Lemma 1. Let $K\geqslant 1$, and let $\|\cdot\|$ be a nonnegative functional on a vector space $L$ satisfying Let $0<\alpha\leqslant 1$ be a solution of the equation $(2K)^{\alpha}=2$. Then for all $n\in\mathbb N$ and $\{x_k\}_{k=1}^n\subset L$, 3.4. The maximal function $\mathcal N$ and $\mathcal H$-spaces Proof. Let $(x,t)\in\mathbf X$ be arbitrary. Raising the obvious inequality to power $p$ and averaging over $y\in B(x,t)$ we obtain
Now let $\{u_n\}$ be a Cauchy sequence in $\mathcal H^p(\mathbf X)$. By (3.7), $\{u_n\}$ converges at each point $(x,t)\in\mathbf X$ to some measurable function $u$ on $\mathbf X$. Let $\{n_k\}$ be an increasing sequence of indices such that
$$
\begin{equation*}
\|u_{n_k}-u_{n_{k+1}}\|_{\mathcal H^p(\mathbf X)}<2^{-(k+1)}.
\end{equation*}
\notag
$$
From the clear equality we obtain
$$
\begin{equation*}
\mathcal N(u-u_{n_k})(x)\leqslant\sum_{j=k}^{\infty}\mathcal N(u_{n_{j+1}}-u_{n_j})(x), \qquad x\in X.
\end{equation*}
\notag
$$
Hence, by the choice of $\{n_k\}$ the sequence $\{u_{n_k}\}$ converges to $u$ in $\mathcal H^p(\mathbf X)$, and therefore the whole sequence $\{u_n\}$ has the same property. Lemma 2 is proved. In what follows, we frequently use the following notation in proofs:
$$
\begin{equation}
E[\lambda]=\{x\in X\colon \mathcal N u(x)>\lambda\},\qquad\lambda>0,
\end{equation}
\tag{3.8}
$$
and
$$
\begin{equation}
\mathcal T(E):=\biggl(\bigcup_{x\notin E}D(x)\biggr)^{\mathrm c},\qquad E\subset X;
\end{equation}
\tag{3.9}
$$
here $E^{\mathrm c}$ is the complement of $E$. Lemma 3. Let $u\colon \mathbf X\to\mathbb{C}$, $E\subset X$ and $\lambda>0$. Then (1) $\widehat{\mathcal T(E)}=E$; (2) $|u(x,t)|\leqslant\lambda$ for $(x,t)\notin\mathcal T(E[\lambda)])$; (3) $\{(x,t)\in\mathbf X\colon |u(x,t)|>\lambda\}\subset\mathcal T(E[\lambda)])$. Proof. (1) If $x\in\widehat{\mathcal T(E)}$, then $D(x)\cap\mathcal T(E)\ne\varnothing$, but if $x\notin E$, then $D(x)\cap\mathcal T(E)=\varnothing$ since Hence $\widehat{\mathcal T(E)}\subset E$. Conversely, if $x\in E$, but $x\notin\widehat{\mathcal T(E)}$, then $D(x)\cap \mathcal T(E)=\varnothing$, that is, $D(x)\subset [\mathcal T(E)]^{\mathrm c}$.
(2) If $(x,t)\notin\mathcal T(E[\lambda)])$, then and $(x,t)\in D(y)$ for some $y\notin E[\lambda]$.Inclusion (3) is immediate from (2). This proves the lemma. A splitting of the function $u=u_\lambda+u^\lambda$ into the parts
$$
\begin{equation}
u^\lambda:= \begin{cases} u & \text{if }|u|>\lambda, \\ 0 & \text{if }|u|\leqslant\lambda \end{cases} \quad\text{and} \quad u_\lambda:= \begin{cases} 0 & \text{if }|u|>\lambda, \\ u & \text{if }|u|\leqslant\lambda, \end{cases}
\end{equation}
\tag{3.10}
$$
is always present in the proofs of various variants of Marcinkiewicz’s interpolation theorem. The following lemma, albeit simple, is important because it shows how the nontangential maximal function $\mathcal N u$ is transformed with the splitting (3.10). Lemma 4. If $u\colon \mathbf X\to\mathbb{C}$ and $\lambda>0$, then and Proof. If $\mathcal N u(x)>\lambda$, then $\mathcal N u(x):=\mathcal N u^\lambda(x)$, because the supremum in the definitions of $\mathcal N u(x)$ and $\mathcal N u^\lambda(x)$ is taken over the same set and $\mathcal N u_\lambda(x)\leqslant\lambda$, because $|u_\lambda(y,t)|\leqslant\lambda$ for all $(y,t)\in\mathbf X$.
If $\mathcal N u(x)\leqslant\lambda$, then $|u(y,t)|\leqslant \lambda$ for $(y,t)\in D(x)$, and for these $(y,t)$, we have $u^\lambda(y,t)=0$ and $u_\lambda(y,t)=u(y,t)$. This proves the lemma. 3.5. Kalton’s lemmaThe following lemma, which is a key ingredient in the proof of Theorem 3, is similar to Lemma 1.4.20 in [9], which (as noted in [9], p. 74) was proposed by Kalton. Lemma 5. Let $0<p<\infty$ and $0<q\leqslant\infty$, and let $T$ be a quasilinear continuous operator $\mathcal H^{p}(\mathbf X)\to L^0(Y)$ satisfying the following condition: there exists a positive constant $M$ such that for each measurable set $A\subset \mathbf X$ such that $\mu(\widehat{A})<\infty$. Next, let $\alpha_0$ satisfy
$$
\begin{equation}
0<\alpha_0<\min\biggl\{q,\frac{\ln 2}{\ln (2K_1)}\biggr\}.
\end{equation}
\tag{3.12}
$$
Then for each $0<\alpha\leqslant\alpha_0$ and any function $u\in \mathcal H^{p,\alpha}(\mathbf X)$ in the domain of $T$, ($\lesssim$ depends on $K_1$, $p$, $q$, $\alpha$). Proof. First we note that if $\alpha_1$ is a solution of the equation $(2K_1)^{\alpha_1}=2$, then by Lemma 1, for all $0<\alpha\leqslant\alpha_1$ and $\{v_k\}_{k=1}^{m}\subset \mathcal H^p(\mathbf X)$, we have the pointwise inequality
Now let $0<\alpha_0\leqslant\alpha_1$ be a number such that $\alpha_0<q$. Then $q/\alpha>1$ for each $0<\alpha\leqslant\alpha_0$, and so the quasinormed Lorentz space $L^{q/\alpha,\infty}(Y)$ is normed (see § 3.2): the norm
$$
\begin{equation*}
\|f\|_{L^{q/\alpha,\infty}(Y)}^*:=\sup\biggl\{[\nu(E)]^{q/\alpha-1}\int_E|f|\,d\nu\colon 0<\nu(E)<\infty\biggr\}
\end{equation*}
\notag
$$
satisfies inequalities (3.4) for $p=q/\alpha$.
Let us now prove the following key fact: there exists a constant $C=C(q,\alpha)$ such that for each nonnegative bounded function $u\in\mathcal H^p(\mathbf X)$ and any measurable set $A\subset\mathbf X$ satisfying $\mu(\widehat{A})<\infty$,
$$
\begin{equation}
\|T(u\chi_A)\|_{\mathcal H^{q,\infty}(Y)}\leqslant C(q,\alpha)M[\mu(\widehat{A})]^{1/p}\|u\|_{L^{\infty}(\mathbf X)}.
\end{equation}
\tag{3.15}
$$
It can be assumed without loss of generality that $\|u\|_{L^{\infty}(\mathbf X)}=1$. The dyadic decomposition of $u\chi_A$ is as follows: where $d_j(x,t)$ is 0 or 1. Setting $A_j:=\{(x,t)\in A\colon d_j(x,t)=1\}$ we can write the decomposition of $u$ asThe space $\mathcal H^p(\mathbf X)$ is complete (see Lemma 2), and so the series (3.16) converges in $\mathcal H^p(\mathbf X)$ because
$$
\begin{equation*}
\|\chi_{A_j}\|_{\mathcal H^p(\mathbf X)}^p=\mu(\widehat{A_j})\leqslant\mu(\widehat{A}).
\end{equation*}
\notag
$$
So, if is the sequence of partial sums of series (3.16), then by the continuity of the operator $T$ the sequence $\{T(u\chi_A-u_m)\}$ converges to zero in measure on $A$, and some subsequence $\{T(u\chi_A- u_{m_k})\}$ of it converges to zero almost everywhere.
Since the operator $T$ is quasilinear (see (1.2) and (1.3)), it follows from (3.14) that
$$
\begin{equation*}
\begin{aligned} \, |T(u\chi_A)| &\leqslant K_1(|Tu_{m_k}|+|T(u\chi_A-u_{m_k}|) \\ &\leqslant K_1\biggl(4\sum_{j=1}^{m_k}|2^{-j}T\chi_{A_j}|^{\alpha}\biggr)^{1/\alpha}+K_1|T(u\chi_A-u_{m_k})|. \end{aligned}
\end{equation*}
\notag
$$
The second term on the right converges to zero almost everywhere as $k\to\infty$, and so we have the pointwise inequality
$$
\begin{equation}
|T(u\chi_A)|\leqslant K_1\biggl(4\sum_{j=1}^{\infty}|2^{-j}T\chi_{A_j}|^{\alpha}\biggr)^{1/\alpha}.
\end{equation}
\tag{3.17}
$$
Now using (3.17), (3.5), (3.4) for $r=q/\alpha$ and (3.11), we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \|T(u\chi_A)\|_{L^{q,\infty}(Y)} &\leqslant 4^{1/\alpha}K_1\biggl\| \biggl(\sum_{j=1}^{\infty}|2^{-j}T\chi_{A_j}|^{\alpha}\biggr)^{1/\alpha}\biggr\|_{L^{q,\infty}(Y)} \\ \notag &=4^{1/\alpha}K_1\biggl\|\sum_{j=1}^{\infty}|2^{-j}T\chi_{A_j}|^{\alpha} \biggr\|_{L^{q/\alpha,\infty}(Y)}^{1/\alpha} \\ \notag &\leqslant K_1\biggl(4\biggl\|\sum_{j=1}^{\infty}|2^{-j}T\chi_{A_j}|^{\alpha} \biggr\|_{L^{q/\alpha,\infty}(Y)}^*\biggr)^{1/\alpha} \\ \notag &\leqslant K_1\biggl(4\sum_{j=1}^{\infty}\|\,|2^{-j}T\chi_{A_j} |^{\alpha}\|_{L^{q/\alpha,\infty}(Y)}^*\biggr)^{1/\alpha} \\ \notag &\leqslant K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha} \biggl(\sum_{j=1}^{\infty} \|\,|2^{-j}T\chi_{A_j}|^{\alpha}\|_{L^{q/\alpha,\infty}(Y)}\biggr)^{1/\alpha} \\ \notag &=K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha} \biggl(\sum_{j=1}^{\infty} \|2^{-j}T\chi_{A_j}\|_{L^{q,\infty}(Y)}^{\alpha}\biggr)^{1/\alpha} \\ \notag &\leqslant K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha}M \biggl(\sum_{j=1}^{\infty}2^{-j\alpha}[\mu(\widehat{A_j})]^{\alpha/p}\biggr)^{1/\alpha} \\ &\leqslant K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha}M(2^{\alpha}-1)^{-1/\alpha} [\mu(\widehat{A})]^{1/p}\|u\|_{L^{\infty}(\mathbf X)}, \end{aligned}
\end{equation}
\tag{3.18}
$$
which proves inequality (3.15) with the constant
$$
\begin{equation*}
C(q,\alpha)=K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha}(2^{\alpha}-1)^{-1/\alpha}.
\end{equation*}
\notag
$$
Let us now verify (3.13) for $u\in\mathcal H^{p}(\mathbf X)$. Using the notation (3.8) and (3.9), consider the sets
$$
\begin{equation*}
\Delta_n=\mathcal T(E[(\mathcal N u)^*(2^{n})])\setminus\mathcal T(E[(\mathcal N u)^*(2^{n+1})]),\qquad n\in\mathbb Z.
\end{equation*}
\notag
$$
It is clear that
$$
\begin{equation*}
\mathbf X=\bigcup_{n\in\mathbb Z}\Delta_n\quad\text{and} \quad \Delta_n\cap \Delta_m=\varnothing.
\end{equation*}
\notag
$$
By Lemma 3
$$
\begin{equation}
u(x,t)\leqslant(\mathcal N u)^*(2^{n}) \quad\text{for } (x,t)\notin\mathcal T(E[(\mathcal N u)^*(2^{n})])
\end{equation}
\tag{3.19}
$$
and
$$
\begin{equation}
\mu(\widehat{\Delta_n})\leqslant\mu(E[(\mathcal N u)^*(2^{n})])=2^{n}
\end{equation}
\tag{3.20}
$$
because
$$
\begin{equation*}
\Delta_n\subset\mathcal T(E[(\mathcal N u)^*(2^{n})])\quad\Longrightarrow\quad\widehat{\Delta_n}\subset E[(\mathcal N u)^*(2^{n})]
\end{equation*}
\notag
$$
and since the functions $\mathcal N u$ and $(\mathcal N u)^*$ are equimeasurable.
We expand $u$ in a series: We claim that this series converges to $u$ in $\mathcal H^p(\mathbf X)$. To prove this we set for brevity
$$
\begin{equation*}
\mathcal T_n:=\mathcal T(E[(\mathcal N u)^*(2^{n})]),\qquad n\in\mathbb N.
\end{equation*}
\notag
$$
Hence $\Delta_n=\mathcal T_n\setminus\mathcal T_{n+1}$ and, for all $N,M\in\mathbb N$,
$$
\begin{equation*}
\mathbf X\setminus\bigcup_{n=-M}^{N-1}\Delta_n=\mathcal T_N\cup(\mathbf X\setminus\mathcal T_{-M}).
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
\biggl|u-\sum_{n=-M}^{N-1}u\chi_{\Delta_n}\biggr|=\bigl|u\chi_{\mathcal T_N}+u\chi_{(\mathbf X\setminus\mathcal T_{-M}})\bigr|=|u\chi_{\mathcal T_N}|+|u\chi_{(\mathbf X\setminus\mathcal T_{-M}})|.
\end{equation}
\tag{3.22}
$$
By Lemma 3 and Lebesgue’s dominated convergence theorem,
$$
\begin{equation*}
\|u\chi_{\mathcal T_N}\|_{\mathcal H^p(\mathbf X)}^p=\int_{\{\mathcal N u>2^N\}}(\mathcal N u)^p\,d\mu\to0,\qquad N\to\infty.
\end{equation*}
\notag
$$
Next, from Lemma 3 we obtain
$$
\begin{equation*}
\mathcal N(u\chi_{(\mathbf X\setminus\mathcal T_{-M})})=\mathcal N u \quad\text{on the set}\ \{\mathcal N u\leqslant2^{-M}\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal N(u\chi_{(\mathbf X\setminus\mathcal T_{-M})})\leqslant 2^{-M} \quad\text{on the set } \{\mathcal N u>2^{-M}\}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, & \|u\chi_{(\mathbf X\setminus\mathcal T_{-M})}\|_{\mathcal H^p(\mathbf X)}^p=\int_{\{\mathcal N u\leqslant2^{-M}\}}(\mathcal N u)^p\,d\mu+\int_{\{\mathcal N u>2^{-M}\}}(\mathcal N u)^p\,d\mu \\ &\qquad \leqslant\int_{\{\mathcal N u\leqslant2^{-M}\}}(\mathcal N u)^p\,d\mu+2^{-Mp}\mu(\{\mathcal N u>2^{-M}\})\to0,\qquad M\to\infty. \end{aligned}
\end{equation*}
\notag
$$
Consequently, the $\mathcal H^p(\mathbf X)$-quasinorms of both terms on the right in (3.22) converge to zero as $N,M\to\infty$, which proves the $\mathcal H^p(\mathbf X)$-convergence of the series (3.21).
The operator $T$ is continuous, and so, proceeding as in the proof of (3.17), we get that
$$
\begin{equation*}
|Tu|\leqslant K_1\biggl(4\sum_{n\in\mathbb Z}|T(u\chi_{\Delta_n})|^{\alpha}\biggr)^{1/\alpha}.
\end{equation*}
\notag
$$
Using this inequality and also (3.15), (3.19) and (3.20), and repeating the proof of (3.18) we obtain
$$
\begin{equation*}
\begin{aligned} \, & \|Tu\|_{L^{q,\infty}(Y)}\leqslant K_1\biggl(\frac{4q}{q-\alpha}\biggr)^{1/\alpha} \biggl(\sum_{n\in\mathbb Z}\|T(u\chi_{\Delta_n})\|_{L^{q,\infty}(Y)}^{\alpha}\biggr)^{1/\alpha} \\ &\qquad \leqslant4K_1^2\biggl(\frac{4q}{q-\alpha}\biggr)^{2/\alpha}(2^{\alpha}-1)^{-1/\alpha}M \biggl(\sum_{n\in\mathbb Z}[\mu(\widehat{\Delta_n})]^{\alpha/p}\|u\chi_{\Delta_n}\|_{L^\infty(\mathbf X)}^{\alpha}\biggr)^{1/\alpha} \\ &\qquad \leqslant 4K_1^2\biggl(\frac{4q}{q-\alpha}\biggr)^{2/\alpha}(2^{\alpha}-1)^{-1/\alpha}M \biggl(\sum_{n\in\mathbb Z}[(\mathcal N u)^*(2^n)]^{\alpha}2^{n\alpha/p}\biggr)^{1/\alpha}. \end{aligned}
\end{equation*}
\notag
$$
To estimate the last sum it suffices to invoke (3.6). Thus, we have proved the conclusion of Lemma 5 for nonnegative functions $u\in\mathcal H^p(\mathbf X)$. We can represent any complex-valued function $u$ as $u=u_1-u_2+i(v_1-v_2)$, where the nonnegative functions $u_j$ and $v_j$, $j=1,2$, are defined by
$$
\begin{equation*}
u_1=\max\{\operatorname{Re} u,0\},\qquad u_2=u_1-\operatorname{Re} u,\qquad v_1=\max\{\operatorname{Im} u,0\},\qquad v_2=v_1-\operatorname{Im} u.
\end{equation*}
\notag
$$
Applying our results to each of the functions $u_j$ and $v_j$, $j=1,2$, since the operator $T$ is quasilinear, we complete the proof of Lemma 5. § 4. Proofs of the main results4.1. Proof of Theorem 3We proceed as in the proof of Theorem 2 (see [9], § 1.4.4). Let $0<p_0<p_1<\infty$ (the case $p_1=\infty$ is considered separately). First, using conditions (1.10) and (1.11) and Lemma 5, we find that
$$
\begin{equation}
\|Tu\|_{L^{q_0,\infty}(Y)}\leqslant C_0M_0\|\mathcal N u\|_{L^{p_0,\beta}(X)}, \qquad u\in \mathcal H^{p_0,\beta}(X),
\end{equation}
\tag{4.1}
$$
and
$$
\begin{equation}
\|Tu\|_{L^{q_1,\infty}(Y)}\leqslant C_1M_1\|\mathcal N u\|_{L^{p_1,\beta}(X)}, \qquad u\in \mathcal H^{p_1,\beta}(X),
\end{equation}
\tag{4.2}
$$
where the constants
$$
\begin{equation*}
C_0:=C(K_1,p_0,q_0,\beta)\quad\text{and} \quad C_1:=C(K_1,p_1,q_1,\beta)
\end{equation*}
\notag
$$
are determined from inequality (3.13) for
$$
\begin{equation*}
\beta:=\frac{1}{2}\min\biggl\{q_0,q_1,2r,\frac{\ln 2}{\ln 2K_1}\biggr\}.
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
\gamma:=\frac{1/q_0-1/q}{1/p_0-1/p}=\frac{1/q-1/q_1}{1/p-1/p_1}
\end{equation*}
\notag
$$
and for any $t>0$ we represent $u\in\mathcal H^{p,r}(\mathbf X)$ as the sum $u=u^\lambda+u_\lambda$ (see (3.10)), where $\lambda:=(\mathcal N u)^*(\delta t^{\gamma})$ and $\delta>0$ is a number to be chosen below from some optimization considerations. From (3.10), Lemma 4, and the definition of an equimeasurable rearrangement we obtain
$$
\begin{equation}
\begin{cases} (\mathcal N u^\lambda)^*(s)\leqslant(\mathcal N u)^*(s) &\text{for }0<s\leqslant\delta t^\gamma, \\ (\mathcal N u^\lambda)^*(s)=0 & \text{for } s>\delta t^\gamma, \\ (\mathcal N u_\lambda)^*(s)=(\mathcal N u)^*(\delta t^\gamma) & \text{for } 0<s\leqslant\delta t^\gamma, \\ (\mathcal N u_\lambda)^*(s)\leqslant (\mathcal N u)^*(s) & \text{for } s>\delta t^\gamma. \end{cases}
\end{equation}
\tag{4.3}
$$
In particular, by (4.3) we have $u^\lambda\in\mathcal H^{p_0,\beta}(\mathbf X)$ and $u_\lambda\in\mathcal H^{p_1,\beta}(\mathbf X)$ for each $t>0$. Let us now estimate $\|Tu\|_{L^{q,r}(Y)}$. Since the operator $T$ is quasilinear, we have
$$
\begin{equation}
\begin{aligned} \, \notag &\|Tu\|_{L^{q,r}(Y)}=\|t^{1/q}(Tu)^*(t)\|_{L^r(dt/t)}\leqslant K_1\|t^{1/q}\bigl[(Tu^\lambda)+(Tu_\lambda)\bigr]^*(t)\|_{L^r(dt/t)} \\ \notag &\qquad \leqslant K_1\|t^{1/q}\biggl[(Tu^\lambda)^*\biggl(\frac t2\biggr)+(Tu_\lambda)^*\biggl(\frac t2\biggr)\biggr]\|_{L^r(dt/t)} \\ &\qquad \leqslant2^{1/r}K_1\biggl(\biggl\|t^{1/q}(Tu^\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)} +\biggl\|t^{1/q}(Tu_\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)}\biggr). \end{aligned}
\end{equation}
\tag{4.4}
$$
Let us estimate separately each term on the right in (4.4). From inequalities (4.1) and (4.2) we obtain
$$
\begin{equation}
t^{1/q_0}(Tu^\lambda)^*\biggl(\frac{t}2\biggr)\leqslant 2^{1/q_0}\sup_{s>0}\{s^{1/q_0}(Tu^\lambda)^*(s)\}\leqslant2^{1/q_0}C_0M_0 \|\mathcal N u^\lambda\|_{L^{p_0,\beta}(\mathbf X)}
\end{equation}
\tag{4.5}
$$
and
$$
\begin{equation}
t^{1/q_1}(Tu_\lambda)^*\biggl(\frac{t}2\biggr)\leqslant 2^{1/q_1}\sup_{s>0}\{s^{1/q_1}(Tu_\lambda)^*(s)\}\leqslant2^{1/q_1}C_1M_1 \|\mathcal N u_\lambda\|_{L^{p_1,\beta}(\mathbf X)},
\end{equation}
\tag{4.6}
$$
respectively. Using (4.5) and since we have the estimate
$$
\begin{equation*}
\begin{aligned} \, \biggl\|t^{1/q}(Tu^\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)} &=\biggl\|t^{1/q-1/q_0}t^{1/q_0}(Tu^\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)} \\ &\leqslant2^{1/q_0}C_0M_0\|t^{1/q-1/q_0}\|\mathcal N u^\lambda\|_{L^{p_0,\beta}(X)}\|_{L^r(dt/t)} \\ &=2^{1/q_0}C_0M_0\|t^{\gamma(1/p_0-1/p)}\|\mathcal N u^\lambda\|_{L^{p_0,\beta}(X)}\|_{L^r(dt/t)}. \end{aligned}
\end{equation*}
\notag
$$
Changing the variable to $\tau:=\delta t^\gamma$, employing the first two relations in (4.3) and applying Hardy’s inequality (3.1) for $p=r/\beta\geqslant 1$ and $b=r(1/p_0-1/p)$, we continue the estimate as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl\|t^{1/q}(Tu^\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)} \\ \notag &\qquad \leqslant2^{1/q_0}C_0M_0\frac{\delta^{1/p_0-1/p}}{\gamma^{1/r}} \biggl\|\tau^{1/p_0-1/p}\biggl[\int_0^{\tau}[(\mathcal N u)^*(s)]^{\beta}s^{\beta/p_0}\,\frac{ds}{s}\biggr]^{1/\beta}\biggr\|_{L^r(d\tau/\tau)} \\ \notag &\qquad \leqslant2^{1/q_0}C_0M_0\frac{\delta^{1/p_0-1/p}}{\gamma^{1/r}} \biggl[\frac{r/\beta}{r(1/p_0-1/p)}\biggr]^{1/\beta} \\ \notag &\qquad\qquad \times\biggl(\int_0^{\infty}[s^{1/p_0}(\mathcal N u)^*(s)]^rs^{-r(1/p_0-1/p)}\,\frac{ds}{s}\biggr)^{1/r} \\ &\qquad =2^{1/q_0}C_0M_0\frac{\delta^{1/p_0-1/p}}{\beta^{1/\beta}\gamma^{1/r}(1/p_0-1/p)^{1/\beta}} \|\mathcal N u\|_{L^{p,r}(X)}. \end{aligned}
\end{equation}
\tag{4.7}
$$
The second term on the right in (4.4) is estimated similarly: first, from inequality (4.6) and the equalities we obtain
$$
\begin{equation*}
\biggl\|t^{1/q}(Tu_\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)}\leqslant2^{1/q_1}C_1M_1 \|t^{\gamma(1/p-1/p_1)}\|\mathcal N u_\lambda\|_{L^{p_1,\beta}(X)}\|_{L^r(dt/t)}.
\end{equation*}
\notag
$$
Changing the variable to $\tau:=\delta t^\gamma$, using the last two inequalities in (4.3) and applying Hardy’s inequality (3.2) for
$$
\begin{equation*}
p=\frac{r}{\beta}\geqslant 1\quad\text{and} \quad b=r\biggl(\frac 1p-\frac{1}{p_1}\biggr),
\end{equation*}
\notag
$$
we get that
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl\|t^{1/q}(Tu_\lambda)^*\biggl(\frac t2\biggr)\biggr\|_{L^r(dt/t)} \\ &\qquad \leqslant2^{1/q_1}C_1M_1\frac{2^{1/\beta}\delta^{1/p-1/p_1}}{\beta^{1/\beta}\gamma^{1/r}} \biggl\{p_1^{1/\beta}+\biggl(\frac 1p-\frac{1}{p_1}\biggr)^{-1/\beta}\biggr\} \|\mathcal N u\|_{L^{p,r}(X)}. \end{aligned}
\end{equation}
\tag{4.8}
$$
Substituting estimates (4.7) and (4.8) into the right-hand side of (4.4) we arrive at the inequality
$$
\begin{equation*}
\|Tu\|_{L^{q,r}(Y)}\leqslant C\bigl(M_0\delta^{1/p_0-1/p}+M_1\delta^{1/p-1/p_1}\bigr) \|\mathcal N u\|_{L^{p,r}(X)},
\end{equation*}
\notag
$$
where $C=C(K_1,p_0,p_1,q_0,q_1,r,\theta)$ is a positive constant depending only on the above parameters. Now we choose $\delta>0$ so as to minimize the right-hand side of the last inequality. This is equivalent to saying that Hence
$$
\begin{equation*}
\delta=\biggl(\frac{M_0}{M_1}\biggr)^{1/(1/p_1-1/p_0)}.
\end{equation*}
\notag
$$
This proves inequality (2.6) in the case $0<p_1<\infty$. The case $p_1=\infty$ can be reduced to the one just considered as follows. We set $p^*\in(p,\infty)$ and choose $\theta^*\in(0,1)$ so as to have
$$
\begin{equation*}
\frac{1}{p^*}=\frac{1-\theta^*}{p_0}+\frac{\theta^*}{\infty}; \quad\text{we also set}\ \frac{1}{q^*}:=\frac{1-\theta^*}{q_0}+\frac{\theta^*}{q_1}.
\end{equation*}
\notag
$$
Let $g\in L^0(Y)$, and let $\nu_{g}(\lambda):=\nu\{x\in X\colon |g(x)|>\lambda\}$ be the distribution function of $g$. The following inequality is clear:
$$
\begin{equation*}
\lambda[\nu_g(\lambda)]^{1/q^*} =(\lambda[\nu_g(\lambda)]^{1/q_0})^{1-\theta^*}(\lambda[\nu_g(\lambda)]^{1/q_1})^{\theta^*} \leqslant\|g\|_{L^{q_0,\infty}(Y)}^{1-\theta^*}\|g\|_{L^{q_1,\infty}(Y)}^{\theta^*}.
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation*}
\|g\|_{L^{q^*,\infty}(Y)}\leqslant\|g\|_{L^{q_0,\infty}(Y)}^{1-\theta^*} \|g\|_{L^{q_1,\infty}(Y)}^{\theta^*}.
\end{equation*}
\notag
$$
Now from the main conditions (1.10) and (1.11) we obtain
$$
\begin{equation*}
\|T\chi_A\|_{L^{q^*,\infty}(Y)}\leqslant M_0^{1-\theta^*}M_1^{\theta^*}[\mu(\widehat{A})]^{1/p^*}.
\end{equation*}
\notag
$$
This means that the operator $T$ is of restricted weak type $(p^*,q^*)$. An application of what has been proved for the pairs $(p_0,q_0)$ and $(p^*,q^*)$ shows that
$$
\begin{equation*}
\|Tu\|_{L^{q,r}(X)}\leqslant C M_0^{1-\omega}(M_0^{1-\theta^*}M_1^{\theta^*})^{\omega}\|u\|_{\mathcal H^{p,r}(\mathbf X)}, \qquad u\in S_0(\mathbf X),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\frac{1}{p}=\frac{1-\omega}{p_0}+\frac{\omega}{p^*}\quad\text{and} \quad \frac{1}{q}=\frac{1-\omega}{q_0}+\frac{\omega}{q_1}.
\end{equation*}
\notag
$$
In view of the relations between the parameters we have $\theta=\omega\theta^*$, and therefore the last inequality coincides with (2.6). 4.2. Proof of Theorem 4This result is a direct consequence of Theorem 3, since by the substitution of the characteristic functions of sets into inequalities (2.8) and (2.9) we obtain (2.4) and (2.5), respectively. The continuity of the operator $T\colon \mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)\to L^0(Y)$ is secured by (2.8) and (2.9). 4.3. Proof of Theorem 5For a proof it suffices to apply Theorem 3 to the operator
$$
\begin{equation*}
\mathcal N_Y\circ T\colon \mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)\to L^0(\mathbf Y),
\end{equation*}
\notag
$$
which is quasilinear and continuous.
§ 5. Applications5.1. The Carleson–Duren–Hörmander embedding theoremA Borel measure $\nu$ on $\mathbf X$ is said to be a Carleson measure of order $\alpha\!>\!0$ (written ${\nu\mkern-1mu\!\in\!\mkern-1mu CM_{\alpha}\mkern-1mu(\mathbf X)}$) if
$$
\begin{equation}
\sup_{B\subset X}\nu(\mathcal T(B))[\mu(B)]^{-\alpha}:=\|\mu\|_{CM_{\alpha}(\mathbf X)}<\infty,
\end{equation}
\tag{5.1}
$$
where the supremum is taken over all balls $B\subset X$ and the set $\mathcal T(B)$ was defined in (3.9) (see, for example, Ch. 3, § 1.4 in [28]) A measure $\mu$ on $X$ is said to satisfy the doubling condition if there exists a positive number $K_3=K_3(\mu)$ such that
$$
\begin{equation}
\mu(B(x,2t))\leqslant K_3\mu(B(x,t)), \qquad x\in X, \quad t>0.
\end{equation}
\tag{5.2}
$$
Lemma 6. Let the measure $\mu$ on $X$ satisfy the doubling condition (5.2), and let $\nu\in CM_{\alpha}(\mathbf X)$ for some $\alpha\geqslant1$. Then for each measurable set $A\subset\mathbf X$ such that $\mu(\widehat{A})<\infty$, ($\lesssim$ depends only on $\|\mu\|_{CM_{\alpha}(\mathbf X)}$ and $K_3$ from (5.2)). Proof. Consider the case $t_0=\infty$ (see (2.1)). Let
If $x\in\widehat{A_n}$ (recall that $\widehat{A_n}$ is defined in (2.3)), then there exists a point $(y,t)\in\mathbf X$ such that (see (2.2)). We set $B_x:=B(y,t)\subset\widehat{A_n}$. The family of balls $\{B_x\colon x\in\widehat{A_n}\}$ of uniformly bounded radii covers $\widehat{A_n}$, and this family contains a finite or countable subfamily $\{B_k:=B_{x_k}\}$ such that
$$
\begin{equation*}
B_k\cap B_j=\varnothing\quad\text{and} \quad \widehat{A_n}\subset\bigcup_{k}B^*_k,
\end{equation*}
\notag
$$
where $B^*_k$ is the ball concentric with $B_k$ and of radius $3K_2^2$ times greater than that of $B^*_k$ (see [29], Lemma 3). Now we have
$$
\begin{equation*}
A_n\subset\mathcal T(\widehat{A_n})\subset\bigcup_{k}\mathcal T(B^*_k).
\end{equation*}
\notag
$$
Since $\mu\in CM_{\alpha}(\mathbf X)$, $\alpha\geqslant1$, and using also the doubling condition (5.2), we get that
$$
\begin{equation*}
\nu(A_n)\leqslant\sum_{k}\nu(\mathcal T(B^*_k))\leqslant\sum_{k}[\nu(B^*_k)]^{\alpha}\lesssim\biggl(\sum_{k}\mu(B_k)\biggr)^{\alpha} \lesssim[\mu(\widehat{A_n})]^{\alpha}\lesssim[(\mu(\widehat{A})]^{\alpha}.
\end{equation*}
\notag
$$
Letting $n\to\infty$ here, we arrive at (5.3), since
$$
\begin{equation*}
\lim_{n\to\infty}\nu(A_n)=\nu\biggl(\bigcup_{n=1}^{\infty}A_n\biggr) =\nu(\widehat{A}).
\end{equation*}
\notag
$$
In the case when $t_0<\infty$ the arguments are simpler, since there is no need to introduce the sequence $\{A_n\}$. Lemma 6 is proved. Theorem 6. Let $0<p\leqslant q<\infty$, let the measure $\mu$ on $X$ satisfy the doubling condition, and let $\nu$ be a measure on $\mathbf X$ whose domain contains that of the measure $\mu\times m_1$. Then the following conditions are equivalent: (1) $\nu$ is a Carleson measure of order $q/p$ on $\mathbf X$; (2) inequality (5.3) holds for each measurable set $A\subset\mathbf X$, $\mu(\widehat{A})<\infty$ ($\lesssim$ does not depend on $A$); (3) for all $u\in\mathcal H^p(\mathbf X)$ and $\lambda>0$,
$$
\begin{equation}
\nu\{(x,t)\in\mathbf X\colon |u(x,t)|>\lambda\}\lesssim\biggl(\frac{\|u\|_{\mathcal H^p(\mathbf X)}}{\lambda}\biggr)^q
\end{equation}
\tag{5.4}
$$
($\lesssim$ does not depend on $\lambda$ and $u$); (4) for each function $u\in\mathcal H^p(\mathbf X)$, ($\lesssim$ does not depend on $u$). Proof. We need to verify only the implication (1) $\Rightarrow$ (4). To do this, let $0<p_0<p<p_1<\infty$ and $0<q_0<q<q_1<\infty$ be such that
$$
\begin{equation*}
p_0\leqslant q_0, \qquad p_1\leqslant q_1\quad\text{and} \quad \frac{q_0}{p_0}=\frac{q_1}{p_1}=\frac{q}{p}.
\end{equation*}
\notag
$$
Consider the identity operator $\mathrm{Id}\colon \mathcal H^{p_0}(\mathbf X)+\mathcal H^{p_1}(\mathbf X)\to L^0(\mathbf X)$. Clearly, this operator is linear and continuous. We apply Lemma 6 to each of the pairs $(p_0,q_0)$ and $(p_1,q_1)$ (for these pairs (5.3) clearly corresponds to conditions (2.4) and (2.5) for this operator). Now Theorem 6 follows from Corollary 1. Let us illustrate Theorem 6 with two particular cases. Example 1. For $n\geqslant 1$ let $B^n\subset\mathbb{C}^n$ be the open unit ball in $\mathbb{C}^n$. Let $X=S=\partial B^n\subset\mathbb{C}^n$ be the unit sphere (the boundary of $B^n$) and $\mu=\sigma$ be the surface Lebesgue measure on $S$ normalized by the condition $\sigma(S)=1$. Consider the natural quasimetric The Hardy class $H^p(B^n)$, $p>0$, consists of the holomorphic functions $f\colon {B^n\!\to\!\mathbb{C}}$ with finite norm
$$
\begin{equation*}
\|f\|_{H^p(B^n)}:=\sup_{0\leqslant r<1}\|f_r\|_{L^p(S)},\qquad f_r(\zeta):=f(r\zeta)
\end{equation*}
\notag
$$
(see [20], § 5.6, and [28], Ch. 2, § 1.3). Identifying the punctured ball $B^n\setminus\{0\}$ with $\mathbf X=S\times(0,1)$ via
$$
\begin{equation*}
z\mapsto (x,t), \quad\text{where } x=\frac{z}{|z|}\quad\text{and} \quad t=1-|z|,
\end{equation*}
\notag
$$
we find that $H^p(B^n)$ for $p>0$ is continuously embedded in $\mathcal H^p(S\times(0,1))$ (see, for example, [20], Theorem 5.6.5). Now Theorem 6 implies that if $0<p\leqslant q<\infty$ and $\nu$ is a Carleson measure on $B^n$ of order $q/p$, then
$$
\begin{equation}
\biggl(\int_{B^n}|f|^q\,d\nu\biggr)^{1/q}\lesssim\|\mathcal Nf\|_{L^p(S)}\lesssim\|f\|_{H^p(B^n)}.
\end{equation}
\tag{5.7}
$$
The first inequality here holds for any measurable function $f\colon S\times(0,1)\to\mathbb{C}$, and the second holds for $f\in H^p(B^n)$. The definition of a Carleson measure dates back to Carleson’s papers [30] and [31], where inequalities of the form (5.7) (for $n=1$ and $q=p$) were used to solve important problems on Hardy classes of analytic functions on the unit disc (namely, the problem of interpolation by analytic functions and the corona problem). For more details, see also [32], Chs. 7 and 8. Hörmander [33] extended inequality (5.7) for $q=p$ to the Hardy classes of holomorphic functions on domains in $\mathbb{C}^n$, $n\geqslant1$, with sufficiently smooth boundaries; the diagonal variant of Marcinkiewicz’s interpolation theorem was used in the proof. Duren [34] employed Hörmander’s method in the one-dimensional case with ${0<p\leqslant q<\infty}$ (also see the discussion of [33] and [34] in [28], § 1.4). Example 2. Let $X=\mathbb R^n$, $n\geqslant 1$, let $d(x,y)=|x-y|$ be the Euclidean metric, and $\mu$ be the Lebesgue measure on $\mathbb R^n$, $I=(0,\infty)$. Here, one can take as the Hardy class $H^p(\mathbb R^{n+1}_+)$, $p>0$, for example, the set of harmonic functions $u$ in $\mathbb R^{n+1}_+:=\mathbb R^n\times\mathbb R_+$ whose nontangential maximal function $\mathcal N u$ lies in $L^p(\mathbb R^n)$. The following more general variant of the classes $H^p$ can also be used. Let $\varphi$ be a sufficiently smooth function on $\mathbb R^n$ (for example, $\varphi$ lies in the Schwartz class), and let
$$
\begin{equation*}
\varphi_t(x):=t^{-n}\varphi\biggl(\frac xt\biggr), \qquad t>0.
\end{equation*}
\notag
$$
Let $H^p_{\varphi}(\mathbb R^{n+1}_+)$ be the class of convolutions $u=f\ast \varphi_t$ of tempered distributions $f$ whose nontangential maximal function $\mathcal N(f\ast \varphi_t)$ lies in $L^p(\mathbb R^n)$ (for more details, see [19] or [35], Ch. 2). Then and this embedding is continuous. The possible use of nontangential maximal functions in the proof of Carleson’s embedding theorem for Poisson integrals of $L^p(\mathbb R^n)$-functions, $p>1$, was pointed out in [4], Ch. 7, § 4.4. In this regard we also mention the paper [36], where this idea was used to study Carleson measures on products of the form $X\times(0,\infty)$, where $X$ is a quasimetric space equipped with a measure satisfying the doubling condition (5.2). However, in [36] the functions were additionally assumed to satisfy a Harnack-type inequality. 5.2. Hardy–Littlewood inequalitiesHere we consider some inequalities for $\mathcal H^p(\mathbf X)$-functions, $p>0$, which originate from the Hardy–Littlewood inequalities for analytic functions in Hardy classes in the unit disc in $\mathbb{C}$. We set
$$
\begin{equation*}
M_p(t,u):=\biggl(\int_X|u(y,t)|^p\,d\mu(y)\biggr)^{1/p}.
\end{equation*}
\notag
$$
Theorem 7. Assume that for some $n>0$ and $K_4>0$ the measure $\mu$ satisfies We also assume that $0<p<q\leqslant\infty$ and $p\leqslant l$. Then for each $u\in \mathcal H^p(\mathbf X)$
$$
\begin{equation}
|u(x,t)|\lesssim t^{-n/p}\|\mathcal N u\|_{L^p(X)},\qquad x\in X
\end{equation}
\tag{5.9}
$$
($\lesssim$ depends only on $K_4$ and $p$), ($\lesssim$ depends only on $K_4$, $p$ and $q$), and ($\lesssim$ depends only on $K_4$, $p$, $q$ and $l$). Proof. To verify (5.9) we consider $(x,t)\in\mathbf X$, use the obvious inequality
$$
\begin{equation*}
|u(x,t)|\leqslant\mathcal N u(y), \qquad y\in B(x,t),
\end{equation*}
\notag
$$
raise it to power $p$, average over the ball $B(x,t)$ and use condition (5.8). As a result, we obtain
$$
\begin{equation*}
\begin{aligned} \, M^q_q(t,u) &\leqslant\int_{X}[\mathcal N u(x)]^p|u(x,t)|^{q-p}\,d\mu(x) \\ &\lesssim t^{-n(q-p)/p}\|\mathcal N u\|_{L^p(X)}^{q-p}\|\mathcal N u\|_{L^p(X)}^{p}. \end{aligned}
\end{equation*}
\notag
$$
Next we establish (5.11) for $l=p$ first. To this end, for $0<q<\infty$ we introduce the operator which is defined on the class $u\in L^0(\mathbf X)$.From inequalities (5.10), for $0<p<q$ we obtain
$$
\begin{equation}
Tu(x)\lesssim t^{-1/p}\|\mathcal N u\|_{L^p(X)}, \qquad t\in(0,t_0).
\end{equation}
\tag{5.12}
$$
As a result, for all $0<p<q$,
$$
\begin{equation}
m_1\{t\in(0,t_0)\colon Tu(t)>\lambda\}\lesssim\min\{t_0,\lambda^{-p} \|\mathcal N u\|_{L^p(X)}^p\}, \qquad\lambda>0
\end{equation}
\tag{5.13}
$$
(recall that $m_1$ is the Lebesgue measure on $(0,t_0)$).
In addition, it is clear that $T$ is a quasilinear operator. So we can apply Theorem 4 to $T$, which implies (5.11) for $l=p$. The case $l>p$ is reduced via (5.10) to that considered above:
$$
\begin{equation*}
\begin{aligned} \, &\int_0^{t_0}t^{nl(1/p-1/q)-1}M_q^{l}(t,u)\,dt \\ &\qquad =\int_0^{t_0}\bigl[t^{n(1/p-1/q)}M_q(t,u)\bigr]^{l-p}t^{np(1/p-1/q)-1}M_q^{p}(t,u)\,dt \\ &\qquad \lesssim\|\mathcal N u\|_{L_p(X)}^{l-p} \int_0^{t_0}t^{np(1/p-1/q)-1}M_q^{p}(t,u)\,dt \lesssim\|\mathcal N u\|_{L_p(X)}^{l-p}\|\mathcal N u\|_{L_p(X)}^{p}. \end{aligned}
\end{equation*}
\notag
$$
This proves Theorem 7. We illustrate Theorem 7 by using the spaces $X$ from Examples 1 and 2. For the unit disc $B^1\subset\mathbb{C}$ (see Example 1 with $n=1$) inequalities in Theorem 7 were studied by Hardy and Littlewood (see [37], Theorem 2, [38], Theorems 27 and 31, and [39], Theorems 1 and 11), who proved (5.9)–(5.11) for analytic functions in the Hardy classes $H^p(B^1)$. Flett (see [40], Theorem 1) established inequalities (5.9)–(5.11) for $p\geqslant1$ for Poisson integral-type operators on some locally compact groups, and applied them, in particular, to Poisson integrals in the half-space $\mathbb R^{n+1}$ for $p\geqslant 1$ (see [40], Theorem 2) and also, for $p>0$ (see [40], Theorem 3), to functions $u$ such that some power $u^k$, $k\leqslant p$, is subharmonic. Flett’s arguments depend on the diagonal version of Marcinkiewicz’s interpolation theorem. His idea was subsequently used by Mitchell and Hahn (see [41], Theorem 4), who carried over the one-dimensional Hardy–Littlewood inequalities to the multivariate case of holomorphic functions in the Hardy classes $H^p$ on the unit ball or on a bounded symmetric domain in $\mathbb{C}^n$. All these results are corollaries to Theorem 7. 
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